8 LEBESGUE'S CONVERGENCE THEOREMS 133 12. Let X, Y be two locally compact spaces and TT : X -> Y a proper continuous mapping (Section 1 2.7, Problem 2). Let p, be a positive measure on X, and let v 7r(fi) (Section 13.4, Problem 8). Show that v*(g) = ju*(# o TT) for all functions g ;> 0 on Y. (Consider first the case in which g has compact support, and then use Section 1 2.7, Problem 2(b).) Show that N <= Y is v-negligible if and only if TT'^N) is ju-negligible. 13. Assume that there exists in I = [0, 1] an increasing sequence (Hn) of nonmeasurable sets (with respect to Lebesgue measure A) such that \J Hn = I and A*(Hn) = 0 for all n. n Show that AVinf (1 - <pHn)\ = 0, but that A*(l - <pH|I) Ğ 1 for all n. 14. (a) Let F be a convex function defined on a convex open set A c R" (Section 8.5, Problem 8) and let/* (1 ^ k ^ n) be n integrable functions on X, such that the map- ping xi-ğ(/*(x))isfcSfi takes its values in A, and such that the composite function jc H-> F(/i(*), . . . , fn(x)) is integrable. Show that if p is bounded (1 3.9.1 6) and ju,(X) = 1 , then ) £ F( J/x *&..., J/M dp). (Use the Hahn-Banach theorem applied to the convex set of points (fi,..., /, Ğ) E A xR such that w^>F(f !,...,/)-) Consider in particular the case n = 1, F(f) == e*, and deduce the inequality of the means tftul2 u%* :£ aii/i + a2M2 + ---- h amwm, m where the uk are >0, the afc are ^>0 and V ak = 1 . fcĞi (b) Suppose that fi is bounded and ^(X) = l. Show that if two functions /^O, g ^0 are integrable and such that^ ^ 1, then (J/rf/x)(J^ d^) ^ 1 (use (a)). 15. For each finite sequence (Aj) t $ j ħ of /x- integrable sets, put D(A1,...,An) = ( U Aj)-( 0 Aj). \li72in / Vl^y^n / If not all the Aj are ^-negligible , put o-M(Ai, . . . , An) = ftOXAjL, . . . , An))lfjL(A1 u - - u A.). Show that OV(A!, . . . , An) ^ ^ - E <rM(A, , Aj) if none of the Aj is /^-negligible. (Remark that a point of D(Ai} . . . , An) belongs to at least n - I of the sets D(A* , Aj) (/ <j).) 16. If A is Lebesgue measure on the unit interval I = [0, 1] and if $(I) is the set of non- empty closed subsets of I (Section 3.16, Problem 3), show that the function Ah- >A(A) is not continuous with respect to the Hausdorff distance on 3KI). X, we have fi*(/) = lim /*?(/).